In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results. 2000 Mathematics Subject Classification: 58E05, 70H05, 34C25.

Recently it has been argued, that Poincaré supersymmetric field theories admit an underlying loop space hamiltonian (symplectic) structure. Here shall establish this at the level of a general N = 1 supermultiplet. In particular, we advocate the use of a superloop space introduced in [2], and the necessity of using nonconventional auxiliary fields. As an example we consider the nonlinear σ-model...

We develop a systematic framework for studying target space duality at the classical level. We show that target space duality between manifolds M and M̃ arises because of the existence of a very special symplectic manifold. This manifold locally looks like M×M̃ and admits a double fibration. We analyze the local geometric requirements necessary for target space duality and prove that both manifol...

In this paper the problem of compatibility between a nonlinear connection and other geometric structures on Lie algebroids is studied. The notion of dynamical covariant derivative is introduced and a metric nonlinear connection is found in the more general case of Lie algebroids. We prove that the canonical nonlinear connection induced by a regular Lagrangian on a Lie algebroid is the unique co...

A set of enhanced digital integrators (EDI) with improved accuracy via genetic algorithms are proposed in this paper. By specifying a desired power for the integrator to be sought and the interval for comparison, chromosomes consisting of parameters of the enhanced digital integrator are then searched by the genetic algorithm based on root mean squared (RMS) error between the original integrato...

This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. system of SPDE occurs in study pulse propagation randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that strong order time is $ 1/2 $. then used to has convergence probability almost sure 1/2^{-} $, case cubic nonlinear coupli...

This paper presents an excitatory CMOS power consumption and circuit size, which is ideal for integrator is built using operational amplifier. This circuitry is realized in TSMC 0.18μm CMOS Technology. Switched Capacitor (SC) integrator circuit The SC integrator correlated double

We generalize theWeinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. More specifically we significantly improve a result proved earlier jointly with Tokieda: we remove a strong...

We consider Chern-Simons gauged nonlinear sigma model with boundary which has a manifest bulk diffeomorphism invariance. We find that the Gauss’s law can be solved explicitly when the nonlinear sigma model is defined on the Hermitian symmetric space, and the original bulk theory completely reduces to a boundary nonlinear sigma model with the target space of Hermitian symmetric space. We also st...

Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the Hamiltonian system is refactorized using polynomial symplectic maps. This method is analyzed in detail for the three degree of freedom case. We obtain explicit ...